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# Sums of consecutive integers

Video transcript

The sum of four consecutive
odd integers is 136. What are the four integers? So before even
attempting to tackle it, let's think about
what it means to be a consecutive odd integer, what
four consecutive odd integers could be referring to. So we could start,
let's say, we had 3. The next odd integer-- so we
want consecutive odd integers. So we want the next odd
integer-- would be 5. Then the one after that
would be 7, and the one after that would be 9. These are consecutive
odd integers. Another example-- we
could start at 11. Then the next odd integer is 13. The next one is 15. The next one is 17. The example of
non-consecutive odd integers, if someone went from
3 straight to 7, these are not consecutive. The next odd integer
after 3 is 5, not 7. So these are examples of
consecutive odd integers. These are not
consecutive odd integers. So with that out of
the way, let's actually try to tackle this question. And I encourage you to pause
this video right now and try to tackle this on your own
before I take a go at it. Well as you can imagine,
a little bit of algebra might be handy here. So let's let x be equal to
the smallest of the four. So if x is the smallest of the
four consecutive odd integers, how can we express the
other three in terms of x? We'll just use these
examples right over here. If that is x, then how would
we express this in terms of x? Well, this would be x plus 2. The next odd integer
is going to be the previous odd integer plus 2. You're essentially
skipping the next number, which is going to be even. So if you added 1, you'd
just get to an even number, so you have to add 2 to
get to the next odd one. Well if you add 2 again,
if you add 2 to x plus 2, well now you get to x plus 4. You add 2 to that. You get to x plus 6. And you see that
again right over here. If this was x, then
this is x plus 2. This is x plus 4,
and this is x plus 6. So in general, if x is the
smallest of the integers, we can define the other
three as x plus 2, x plus 4, and x plus 6. And let's take the sum of
them and set that equal to 136 and then solve for x. So the smallest one is x. Now the next smallest one
is going to be x plus 2. Now the one after that's
going to be x plus 4, and then the one after that
is going to be x plus 6. So this is the sum of the
four consecutive odd integers, and they tell us that that
is going to be equal to 136. This is equal to 136. And now we can just
literally solve for x. We have this one unknown here. So let's add these
x terms together. So we have one x, two
x's, three x's, four x's. So we can rewrite
those literally as 4x. And then we have 2 plus 4, which
is 6, plus another 6 is 12. 4x plus 12 is equal to 136. So to solve for x, a
good starting point would be to just to
isolate the x terms on one side of the equation
or try to get rid of this 12. Well to get rid of
that 12, we'd want to subtract 12 from
the left-hand side. But we can't just do it
from the left-hand side. Then this equality
wouldn't hold anymore. If these two things were equal
before subtracting the 12, well then if we want
to keep them equal, if we want the left and
the right to stay equal, we've got to subtract
12 from both sides. So subtracting 12
from both sides gives us, well on the left-hand
side, we're just left with 4x. And on the right-hand side,
we are left with 136 minus 12 is 124. Did I do that right? Yeah, 124. So what is x? Well, we just divide both
sides by 4 to solve for x. And we get-- do that in
the same, original color-- x is equal to 124 divided by 4. So let's see. 100 divided by 4 is 25. 24 divided by 4 is 6. 25 plus 6 is 31. And if you don't feel like
doing that in your head, you could also, of course,
do traditional long division. Goes into 124-- 4
doesn't go into 1. 4 goes into 12 three times. 3 times 4 is 12. You subtract, bring
down the next 4. 4 goes into 4 one time. You get no remainder. So x is equal to 31. So x is the smallest
of the four integers. So this right over
here, x is 31. x plus 2 is going to be 33. x plus 4 is going to be 35. And x plus 6 is going to be 37. So our four consecutive
odd integers are 31, 33, 35, and 37.